Let RS RS Find approximations for EG and Var using Taylor expansions of For any xy the bivariate 64257rst order Taylor expansion about is xy remainder Let EXEY The simplest approximation for XY is then XY The approximation for XY ID: 33549
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Forf(R;S)=R=S,thederivativesaref00RR(R;S)=0;f00RS(R;S)=�S�2;andf00SS(R;S)=2R S3.Specically,when=(R;S),wehavef()=R=S;f00RR()=0;f00RS()=�1 (S)2;andf00SS()=2R (S)3.ThenanimprovedapproximationofE(R=S)isE(R=S)E(f(R;S))R S�Cov(R;S) (S)2+Var(S)R (S)3(11)Bythedenitionofvariance,thevarianceoff(X;Y)isVar(f(X;Y))=En[f(X;Y)�E(f(X;Y))]2o(12)UsingE(f(X;Y))f()(fromabove)Var(f(X;Y))En[f(X;Y)�f()]2o(13)ThenusingtherstorderTaylorexpansionforf(X;Y)expandedaroundVar(f(X;Y))Ehf()+f0x()(X�x)+f0y()(Y�y)�f()i2(14)=Ehf0x()(X�x)+f0y()(Y�y))i2(15)=Enf02x()(X�x)2+2f0x()(X�x)f0y()(Y�y)+f02y()(Y�y)2o(16)=f02x()Var(X)+2f0x()f0y()Cov(X;Y)+f02y()Var(Y)(17)Nowwereturntoourexample:f(R;S)=R=Sexpandedaround=(R;S).Sincef0R=S�1;f0S=�R S2and=(R;S),wenowhavef02R()=1 (S)2;f0R()f0S()=�R (S)3;f02S()=(R)2 (S)4.andsoVar(R=S)1 (S)2Var(R)+2�R (S)3Cov(R;S)+(R)2 (S)4Var(S)(18)=(R)2 (S)2"Var(R) (R)2�2Cov(R;S) RS+Var(S) (S)2#(19)=(R)2 (S)2"2R (R)2�2Cov(R;S) RS+2S (S)2#(20)Reference:Kendall'sAdvancedTheoryofStatistics,Arnold,London,1998,6thEdition,Vol-ume1,byStuart&Ord,p.351.Reference:SurvivalModelsandDataAnalysis,JohnWiley&SonsNY,1980,byElandt-JohnsonandJohnson,p.69.